Approximation of Aperiodic Signals Using Non-Integer Harmonic Series: The Generalized NAFASS Approach

Abstract

In this paper, the non-orthogonal amplitude-frequency analysis of smoothed signals (NAFASS) method) is used to approximate discrete aperiodic signals from various complex systems with the non-integer harmonic series (NIHS). When approximating by the NIHS, there is a problem in determining the dispersion law for harmonic frequencies. In the original version of the NAFASS approach, the frequency dispersion law was determined from a linear-difference equation. However, many complex systems in nature have frequency distributions that differ from the linear law, which is used in the conventional Fourier analysis of periodic signals. This paper proposes a generalization of the NAFASS method for describing aperiodic signals by the NIHS with a frequency distribution that satisfies a recursive formula, which coincides with the local generalized geometric mean (GGM). The methodology of the generalized NAFASS method is demonstrated using descriptions of financial data (prices for metals) and sound data (sounds of insects) as examples. The results show the effectiveness of the generalized NAFASS approach for describing real-world time data. This discovery allows us to propose a new classification scheme for smoothed and aperiodic signals captured as responses and envelopes from various complex systems.